Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{k}{6k(3k - 7)} \div \dfrac{8}{3k - 7} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{k}{6k(3k - 7)} \times \dfrac{3k - 7}{8} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ k \times (3k - 7) } { 6k(3k - 7) \times 8 } $ $ p = \dfrac {k (3k - 7)} {8 \times 6k(3k - 7)} $ $ p = \dfrac{k(3k - 7)}{48k(3k - 7)} $ We can cancel the $3k - 7$ so long as $3k - 7 \neq 0$ Therefore $k \neq \dfrac{7}{3}$ $p = \dfrac{k \cancel{(3k - 7})}{48k \cancel{(3k - 7)}} = \dfrac{k}{48k} = \dfrac{1}{48} $